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Multiway trees & B trees & 2_4 trees

Multiway trees & B trees & 2_4 trees. Go&Ta Chap 10. m-way trees. v 3. v 4. v 5. v 2. v 1. keys>v 5. v 2 < keys<v 3. keys<v 1. . . . . . . Multi-way Search Trees of order m (m-way search trees). Generalization of BSTs Each node has at most m children

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Multiway trees & B trees & 2_4 trees

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  1. Multiway trees & B trees & 2_4 trees Go&Ta Chap 10

  2. m-way trees v3 v4 v5 v2 v1 keys>v5 v2< keys<v3 keys<v1 . . . . . . Multi-way Search Trees of order m (m-way search trees) • Generalization of BSTs • Each node has at most m children • If k is number of values at a node, then node has at most k+1 children (actually exactly m references, but some may be null) • Tree is ordered • BST is a 2-way search tree ADS2 Lecture17

  3. 10 44 22 50 M = 3 Examples A 3-way tree 55 70 3 7 60 68 ADS2 Lecture17

  4. Examples 100 57 50 60 80 63 70 73 58 59 30 35 52 54 61 62 55 56 M = 4 A 4-way tree 4 ADS2 Lecture17

  5. m-way trees Searching in an m-way tree • Similar to that for BST • To search for value x in node (pointed to by) V containing values (v1,…,vk) : • if V=null, we are done (x is not in the tree) • if x<v1, search in V’s left-most subtree • if x>vk, search in V’s right-most subtree, • if x=vi, for some 1ik, we are done (x has been found) • if vixvi+1 for some 1ik-1, search the subtree between vi and vi+1 V v1 v2 …vi vi+1 … vk ADS2 Lecture17

  6. 55 70 3 7 10 44 60 68 22 50 m-way trees Example search for • 68 • 69 • 23 ADS2 Lecture17

  7. m-way trees NOTE: inorder traversal is appropriate/defined

  8. m-way trees Insertion for an m-way tree • Similar to insertion for BST • Remember, for an m-way tree can have at most m-1 values at each node • To add value x, continue as for search until we reach a node (pointed to by V) containing (v1,…,vk) (where k m-1) and can’t continue • If V is not full then add x to V so that values of V remain ordered. • If V is full and x<v1 then the left subtree must be empty, so create • a new (left-most) child for V and place x as its first value. • If V is full and vi < x < vi+1 then the subtree between vi and vi+1 must • be empty, so create a new child for V between vi and vi+1and place x as its first value. • If V is full and x>vkthen the right subtree must be empty, so create • a new (right-most) child for V and place x as its first value ADS2 Lecture17

  9. m-way trees Examples • Create the 4-way tree formed by inserting the values 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 in order ADS2 Lecture17

  10. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16

  11. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 12

  12. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 11,12

  13. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12

  14. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12 14

  15. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12 9 14

  16. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12 3 9 14

  17. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12 2,3 9 14

  18. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12 2,3 9,10 14

  19. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12 2,3,5 9,10 14

  20. M=4 Insert 12, 11, 8, 14, 9, 3, 2, 10, 5, 16 8,11,12 2,3,5 9,10 14,16

  21. m-way trees Node of an m-way tree • Each node contains • Integer size (indicating how many values present) • A reference to the left-most child • A sequence of m-1 value/reference pairs v v v v v v v v v v 3 3 4 4 5 5 2 2 1 1 keys>v keys<v keys<v keys<v . . . . . . v v < keys<v < keys<v . . . . . . 5 5 1 1 2 2 3 3 Inorder Traversal Left subtree traversal, first value, first right subtree traversal, next value, next right subtree traversal etc. ADS2 Lecture17

  22. m-way trees • m could be really big • a node could contain a tree (a bstree or an avl tree) • we might search within node using binary search • nodes might correspond to large regions of disc space • we want to minimise slooooow disc access • think BIG

  23. balanced m-way trees

  24. Balanced m-way trees (B-trees) • Like BSTs, m-way trees can become very unbalanced Here we need to check 5 nodes to find value 55 but only 2 to find value 35 Of particular importance when we want to use trees to process data on secondary storage like disks where access is costly We use a special type of m-way tree (B-tree) which ensures balance: all leaves are at the same depth ADS2 Lecture17

  25. B-Trees Motivation • If we want to store large amounts of data, may need to store it on disk • Number of times we have to access disk to retrieve data becomes important • A disk access is very expensive compared to a typical computer instruction • Number of disk accesses dominates running time • Secondary memory (disk) divided into equal-sized blocks (e.g. 512, 2048, 4096 or 8192 bytes) • Basic I/O operation transfers contents of one disk block to/from main memory • Our goal: to devise m-way search tree which minimises disk access (and exploits disk block read) ADS2 Lecture17

  26. 10 years old! ADS2 Lecture17

  27. A B-trees is: • An m-way search tree designed to conserve space and be reasonably well balanced • Each node still has at most m children but: • Root is either a leaf or has between 2 and m children, and contains at least one value • All nonleaf nodes (except root) have at least • m/2 if even, • at least (m-1)/2 if odd • All leaves are same depth values ADS2 Lecture17

  28. Comparison of B-Trees with binary search trees Comparison with binary search trees: (1) Multi-branched so depth is smaller. Search is faster because there are fewer nodes on a path from root to leaf. (2) Well balanced so the performance of search etc is about optimum. Complexity is logarithmic (like AVL trees..) (3) processing a node takes longer because it has more values. ADS2 Lecture17

  29. 6 11 21 29 22 26 Examples A B-tree of order 5: 30 31 33 3 5 7 9 12 14 17 19 ADS2 Lecture17

  30. Examples 22 44 68 70 50 10 66 55 3 7 A B-tree of order 3: 30 ADS2 Lecture17

  31. Examples 55 70 3 7 10 44 60 68 22 50 Not a B-tree All leaves must be at same depth

  32. Insertion • Like insertion for general m-way search tree, but need to preserve balance condition • To add value x, continue as for search until we reach a node (pointed to by ) V containing (v1,…,vk) (where k m-1) and can’t continue. If we were to add x to V in order. • If V would not overflow, go ahead and add x If V would overflow, add x and split V into 3 parts: Left: first (m-1)/2 values Middle: (m-1)/2 +1 th value Right: last (m-1)/2 values Promote Middle to parent node, with children Left and Right Nb. Assume m is odd. Otherwise Left: first m/2 values Right: last m-2/2 values “Middle”: m/2 +1 th value. ADS2 Lecture17

  33. 71 79 73 75 77 78 81 83 61 64 67 Example To add 74 to this B Tree of order 5 ADS2 Lecture17

  34. Example 71 79 73 75 77 78 81 83 61 64 67 To add 74 to this B-Tree of order 5, would reach node V. Adding 74 would give (ordered) values 73 74 75 77 78 Causing V to overflow. V 34 ADS2 Lecture17

  35. Example 71 79 73 75 77 78 81 83 61 64 67 To add 74 to this B-Tree of order 5, would reach node V. Adding 74 would give (ordered) values 73 74 75 77 78 Causing V to overflow. V split 71 75 79 Promote median to parent node, with children containing 73,74 and 77,78 respectively 73 74 77 78 81 83 61 64 67 35 ADS2 Lecture17

  36. overflow But what if the parent overflows? • If the parent overflows, repeat the procedure (upwards) • If the root overflows, create a new root with Middle its only value and Left and Right as its children ADS2 Lecture17

  37. 12 14 17 19 overflow Example 6 11 21 29 add 18 would cause V to overflow: 12 14 17 18 19 V 22 26 3 5 30 31 33 7 9 ADS2 Lecture17

  38. Example 12 14 17 19 17 overflow 6 11 21 29 add 18 would cause V to overflow: 12 14 17 18 19 V 22 26 3 5 30 31 33 7 9 6 11 21 29 • split v • produce L and R • elevate 17 to parent 7 9 22 26 12 14 18 19 3 5 30 31 33 R L ADS2 Lecture17

  39. Example 12 14 17 19 17 17 21 29 6 11 3 5 7 9 22 26 12 14 18 19 30 31 33 R L overflow 6 11 21 29 add 18 would cause V to overflow: 12 14 17 18 19 V 22 26 3 5 30 31 33 7 9 6 11 21 29 • split v • produce L and R • elevate 17 to parent 7 9 22 26 12 14 18 19 3 5 30 31 33 R L split parent cont. overleaf 39 ADS2 Lecture17

  40. 17 overflow Example contd. 21 29 6 11 3 5 7 9 22 26 12 14 18 19 30 31 33 R L 17 21 29 6 11 3 5 7 9 22 26 12 14 18 19 30 31 33 R L ADS2 Lecture17

  41. 2-4 trees • A B-tree guarantees that insertion, membership and deletion take logarithmic time. • For storing a set it is best to use a B-tree of small order to minimise work at each node (assuming memory resident) • Commonly used are 2-4 B-trees (order 4) In general, a 2-m tree has order m (all non-root nodes have 2,3,..,m children) ADS2 Lecture17

  42. X 2_m Tree An implementation and An example with m=3 A B C

  43. 2_m tree (m=3) X A B C • m = 3 • a node contains at most 2 pieces of data • and then branches 3 ways • a node contains at least one piece of data • and then branches 2 ways • it is a 2-3 tree • m = 4 • a node contains at most 3 pieces of data • an then branches 4 ways • a node contains at least one piece of data • and then branches 2 ways • it is a 2-4 tree

  44. 2_m tree (m=3) X A B C This is null • m = 3 • a node contains at most 2 pieces of data • and then branches 3 ways • a node contains at least one piece of data • and then branches 2 ways • it is a 2-3 tree • m = 4 • a node contains at most 3 pieces of data • an then branches 4 ways • a node contains at least one piece of data • and then branches 2 ways • it is a 2-4 tree

  45. 2_m tree (m=3) X A B C • data (the top row in the picture) an ArrayList • actually contains the stuff that’s in a node

  46. 2_m tree (m=3) X A B C This is null • left (the bottom row in the picture) an ArrayList • pointers to children

  47. 2_m tree (m=3) X A B C This is null • left (the bottom row in the picture) an ArrayList • pointers to children Oops! Should have 4 blocks!

  48. 2_m tree (m=3) X A B C • NOTE: • we do not show parent link • m is the maximum branching factor

  49. 2_m tree (m=3) X A B C • Note: • There are m+1 data and left entries • m data entries used • m+1 left entries used • A null data entry is treated as ∞ • this simplifies overflow

  50. 2_m tree (m=3) X A B C 1 2 4 8 5 6 9 7 Less than 4 Less than 7 Greater than 7 • left.get(i) points to a child with values less that data.get(i) • let n = data.size() • data.get(n-1) == null • left.get(n-1) points to a node with all entries greater than this node • consider data.get(n-1) as infinity

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